Quantitative techniques basics of mathematics permutations and combinations_part ii_30 pages
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Transcript of Quantitative techniques basics of mathematics permutations and combinations_part ii_30 pages
Basic Quantitative Techniques
ABS-Bangalore
1Basic Quantitative Techniques - RVMReddy - ABSApril 8, 2023
Dr. R. Venkatamuni ReddyAssociate Professor
Contact: 09632326277, 080-30938181
2Basic Quantitative Techniques - RVMReddy - ABSApril 8, 2023
Permutations and
Combinations
April 8, 2023 3Basic Quantitative Techniques - RVMReddy - ABS
PermutationsPermutations refers to the different ways in which a
number of a number of objects can be arranged in a different order
Example: Suppose there are two things x and y, they can be arranged in to two different ways i.e,. xy and yx . These two arrangements is called permutation
Similarly x, y and z
xyz, xzy, yxz, yzx, zxy, zyx is 6 arrange permutation
(if we want to have two things only from x,y,z then xy,xz,yz,yx,zx,yz only in this case)April 8, 2023 Basic Quantitative Techniques - RVMReddy - ABS 4
“The word permutation thus refers to the arrangements which can be made by taking some or all of a number of things”
Formulae 1: Finding the number of permutations of ‘n’ dissimilar things taken ‘r’ at a time
n=number of different things given, r=number of different things taken at a time out of different things givenApril 8, 2023 Basic Quantitative Techniques - RVMReddy - ABS 5
Permutations
)1)...(2)(1(),( rnnnnrnP
)!(
!
rn
n
Example 1: There are six boxes and three balls. In how many ways can these three balls be discretely put into these six boxes.
Solution:
Example 2: How many four-letter words can be made using the letters of the word ‘BANGALORE’ and ‘ALLIANCE’
Solution: n=9, r=4 and n=8, r=4 April 8, 2023 Basic Quantitative Techniques - RVMReddy - ABS 6
Permutations
120)!36(
!6),(
rnP
Example 3: How many arrangements are possible of the letters of the words ‘JAIPUR’, ‘BANGALORE’ and ‘ALLIANCE’
Hint: n=6, r=6 and n=9, r=9 and n=8,r=8
April 8, 2023 Basic Quantitative Techniques - RVMReddy - ABS 7
Permutations
Formulae 2: Finding the number of permutations of ‘n’ things taken ‘r’ at a time, given that each of the elements can be repeated once, twice….up to ‘r’ times
Or
‘n’ things taken all at a time of which ‘p’ are alike, ‘q’ others are alike and ‘r’ others alike
Example 1: How many permutations are possible of the letters of the word PROBABILITY when taken all at a time? April 8, 2023 Basic Quantitative Techniques - RVMReddy - ABS 8
Permutations
!!.!.
!),(
rqp
nrnP
Solution: n=11, p=2 ( as letter B is occurring twice in the given word) , and q=2 ( as letter I is occurring twice in the given word)
And all other letters in the given word are different. The required number of permutations is (r is not valid in this)
Example 2: You are given a word “MANAGEMENT” and asked to compute the number of permutations that you can form taking all the letters from this word?
April 8, 2023 Basic Quantitative Techniques - RVMReddy - ABS 9
Permutations
!!.!.
!),(
rqp
nrnP 9979200
!2!.2
!11
Permutations
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Permutation formula proof There are n ways to choose the first element
– n-1 ways to choose the second– n-2 ways to choose the third– …– n-r+1 ways to choose the rth element
By the product rule, that gives us:
P(n,r) = n(n-1)(n-2)…(n-r+1)
April 8, 2023 11Basic Quantitative Techniques - RVMReddy - ABS
Combinations refers to the number of arrangements which can be made from a group of things irrespective of the order
Combinations differ from permutations in that one combination such as xyz may be stated in the form of several permutations just by rearranging the orders as : xyz, xzy, yxz, yzx, zxy, zyx
Note: All of these are one combination but they are six permutations
IMP Note: The number of permutations is always greater than the number of combinations in any given situation since a combination of n different things can be generate n factorial permutations
April 8, 2023 Basic Quantitative Techniques - RVMReddy - ABS 12
Combinations
Formulae 1: The number of r-combinations of a set with n elements, where n is non-negative and 0≤r≤n is:
n= number of different things given
r= number of different things taken at a time out of different things given
April 8, 2023 Basic Quantitative Techniques - RVMReddy - ABS 13
)!(!
!),(
rnr
nrnC
Combinations
Example 1: in how many ways can four persons be chosen out of seven?
n=7, r=4
Example 2: Find the number of combinations of 50 things taking 46 at a time. ANS: 230300
April 8, 2023 Basic Quantitative Techniques - RVMReddy - ABS 14
Combinations
35)!47(!4
!7
)!(!
!),(
rnr
nrnC
Formulae 2: The number of ways in which x+y+z things can be divided into three groups contain x, y, and z things respectively is
Example: In how many ways can 10 books be put to three shelves which can contain 2, 3 and 5 books respectively?
April 8, 2023 Basic Quantitative Techniques - RVMReddy - ABS 15
Combinations
!!.!.
)!(),(
zyx
zyxrnC
2520!5!.3!.2
)!532(),(
rnC
Combinations
16Basic Quantitative Techniques - RVMReddy - ABS
Combinations
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Combinations
How many different poker hands are there (5 cards)?
How many different (initial) blackjack hands are there?
2,598,960!47*1*2*3*4*5
!47*48*49*50*51*52
!47!5
!52
)!552(!5
!52)5,52(
C
1,3261*2
51*52
!50!2
!52
)!252(!2
!52)2,52(
C
April 8, 2023 18Basic Quantitative Techniques - RVMReddy - ABS
Combination formula proof Let C(52,5) be the number of ways to generate
unordered poker hands The number of ordered poker hands is P(52,5) =
311,875,200 The number of ways to order a single poker hand
is P(5,5) = 5! = 120 The total number of unordered poker hands is the
total number of ordered hands divided by the number of ways to order each hand
Thus, C(52,5) = P(52,5)/P(5,5)
April 8, 2023 19Basic Quantitative Techniques - RVMReddy - ABS
Combination formula proof Let C(n,r) be the number of ways to generate
unordered combinations The number of ordered combinations (i.e. r-
permutations) is P(n,r) The number of ways to order a single one of those
r-permutations P(r,r) The total number of unordered combinations is the
total number of ordered combinations (i.e. r-permutations) divided by the number of ways to order each combination
Thus, C(n,r) = P(n,r)/P(r,r)
April 8, 2023 20Basic Quantitative Techniques - RVMReddy - ABS
Combination formula proof
Note that the textbook explains it slightly differently, but it is same proof
)!(!
!
)!/(!
)!/(!
),(
),(),(
rnr
n
rrr
rnn
rrP
rnPrnC
April 8, 2023 21Basic Quantitative Techniques - RVMReddy - ABS
Let n and r be non-negative integers with r ≤ n. Then C(n,r) = C(n,n-r)
Proof:
)!(!
!),(
rnr
nrnC
! )()!(
!),(
rnnrn
nrnnC
)!(!
!
rnr
n
April 8, 2023 22Basic Quantitative Techniques - RVMReddy - ABS
Combination formula proof
Binomial Coefficients
The expression x +y is a binomial expression as it is the sum of two terms.
The expression (x +y)n is called a binomial expression of order n.
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Binomial Coefficients
24Basic Quantitative Techniques - RVMReddy - ABS
Binomial Coefficients
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Binomial Coefficients
Pascal’s Triangle – The number C(n, r) can be obtained by constructing a
triangular array.– The row 0, i.e., the first row of the triangle, contains
the single entry 1. The row 1, i.e., the second row, contains a pair of entries each equal to 1.
– Calculate the nth row of the triangle from the preceding row by the following rules:
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Binomial Coefficients
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Binomial Coefficients The technique known as divide and conquer can be
used to compute C(n, r ). In the divide-and-conquer technique, a problem is
divided into a fixed number, say k, of smaller problems of the same kind.
Typically, k = 2. Each of the smaller problems is then divided into k smaller problems of the same kind, and so on, until the smaller problem is reduced to a case in which the solution is easily obtained.
The solutions of the smaller problems are then put together to obtain the solution of the original problem.April 8, 2023 Basic Quantitative Techniques - RVMReddy - ABS 29
Thank You
30Basic Quantitative Techniques - RVMReddy - ABSApril 8, 2023